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<html>
<head>
<title>
TRIANGLE_NCC_RULE - Newton-Cotes Closed Quadrature Rules for the Triangle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_NCC_RULE <br> Newton-Cotes Closed Quadrature Rules for the Triangle
</h1>
<hr>
<p>
<b>TRIANGLE_NCC_RULE</b>
is a C++ library which
defines Newton-Cotes closed (NCC) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
Newton-Cotes rules have the characteristic that the abscissas
are equally spaced. For a triangle, this refers to spacing
in the unit reference triangle, or in the barycentric coordinate
system. These rules may be mapped to an arbitrary triangle,
and will still be valid.
</p>
<p>
The rules are said to be "closed" when they include points on
the boundary of the triangle.
</p>
<p>
The use of equally spaced abscissas may be important for your
application. That may how your data was collected, for instance.
On the other hand, the use of equally spaced abscissas carries
a few costs. In particular, for a given degree of polynomial
accuracy, there will be rules that achieve this accuracy, but
use fewer abscissas than Newton-Cotes. Moreover, the Newton-Cotes
approach almost always results in negative weights for some
abscissas. This is generally an undesirable feature, particularly
when higher order quadrature rules are being used.
</p>
<p>
(Note that the first rule included in the set is not, strictly
speaking, a Newton-Cotes closed rule; it's just the rule that
uses a single point at the centroid. However, by including
this rule as the first in the set, we have a rule with each
polynomial degree of exactness from 0 to 8.)
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TRIANGLE_NCC_RULE</b> is available in
<a href = "../../c_src/triangle_ncc_rule/triangle_ncc_rule.html">a C version</a> and
<a href = "../../cpp_src/triangle_ncc_rule/triangle_ncc_rule.html">a C++ version</a> and
<a href = "../../f_src/triangle_ncc_rule/triangle_ncc_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_ncc_rule/triangle_ncc_rule.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a C++ library which
defines Grundmann-Moeller quadrature rules
over the interior of a simplex in M dimensions.
</p>
<p>
<a href = "../../cpp_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a C++ library which
defines Newton-Cotes closed quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a C++ library which
sets up a Dunavant quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a C++ library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_lyness_rule/triangle_lyness_rule.html">
TRIANGLE_LYNESS_RULE</a>,
a C++ library which
returns Lyness-Jespersen quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a C++ program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<p>
<a href = "../../cpp_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a C++ library which
defines Newton-Cotes open quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_symq_rule/triangle_symq_rule.html">
TRIANGLE_SYMQ_RULE</a>,
a C++ library which
returns efficient symmetric quadrature rules,
with exactness up to total degree 50,
over the interior of an arbitrary triangle in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../cpp_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a C++ library which
sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a C++ library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Gisela Engeln-Muellges, Frank Uhlig,<br>
Numerical Algorithms with C,<br>
Springer, 1996,<br>
ISBN: 3-540-60530-4,<br>
LC: QA297.E56213.
</li>
<li>
Peter Silvester,<br>
Symmetric Quadrature Formulae for Simplexes,<br>
Mathematics of Computation,<br>
Volume 24, Number 109, January 1970, pages 95-100.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangle_ncc_rule.cpp">triangle_ncc_rule.cpp</a>, the source code.
</li>
<li>
<a href = "triangle_ncc_rule.hpp">triangle_ncc_rule.hpp</a>, the include file.
</li>
<li>
<a href = "triangle_ncc_rule.sh">triangle_ncc_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangle_ncc_rule_prb.cpp">triangle_ncc_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "triangle_ncc_rule_prb.sh">triangle_ncc_rule_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "triangle_ncc_rule_prb_output.txt">triangle_ncc_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FILE_NAME_INC</b> increments a partially numeric file name.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>TRIANGLE_NCC_DEGREE</b> returns the degree of an NCC rule for the triangle.
</li>
<li>
<b>TRIANGLE_NCC_ORDER_NUM</b> returns the order of an NCC rule for the triangle.
</li>
<li>
<b>TRIANGLE_NCC_RULE</b> returns the points and weights of an NCC rule.
</li>
<li>
<b>TRIANGLE_NCC_NUM</b> returns the number of NCC rules available.
</li>
<li>
<b>TRIANGLE_NCC_SUBORDER</b> returns the suborders for an NCC rule.
</li>
<li>
<b>TRIANGLE_NCC_SUBORDER_NUM</b> returns the number of suborders for an NCC rule.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE</b> returns a compressed NCC rule.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_01</b> returns a compressed NCC rule 1.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_02</b> returns a compressed NCC rule 2.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_03</b> returns a compressed NCC rule 3.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_04</b> returns a compressed NCC rule 4.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_05</b> returns a compressed NCC rule 5.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_06</b> returns a compressed NCC rule 6.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_07</b> returns a compressed NCC rule 7.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_08</b> returns a compressed NCC rule 8.
</li>
<li>
<b>TRIANGLE_NCC_SUBRULE_09</b> returns a compressed NCC rule 9.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps T3 reference points to physical points.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA</b> computes the area of a triangle.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 14 June 2014.
</i>
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